Solving The Equation 3/4x + 9/4 = 1/7x - 5 A Step By Step Guide

Introduction: Understanding Linear Equations

In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, linear equations hold a significant place due to their simplicity and wide applicability. Linear equations, characterized by a variable raised to the power of one, appear in numerous real-world scenarios, from calculating distances to determining financial investments. This article delves into the process of solving a specific linear equation: $\frac{3}{4}x + \frac{9}{4} = \frac{1}{7}x - 5$. We will dissect each step involved, providing a clear and concise explanation to empower you with the knowledge to tackle similar problems confidently. Before we dive into the specific equation, let's establish a solid foundation by understanding the core principles of solving linear equations. The primary goal is to isolate the variable, which means getting the variable (in this case, 'x') alone on one side of the equation. We achieve this by performing the same operations on both sides of the equation, ensuring that the equality remains balanced. These operations include addition, subtraction, multiplication, and division. By strategically applying these operations, we can systematically eliminate terms and coefficients surrounding the variable, ultimately leading to the solution. In the following sections, we will apply these principles to the given equation, demonstrating each step with clarity and precision. Prepare to embark on a journey of mathematical discovery, where we unravel the solution to this equation and equip you with the skills to conquer other linear equations that may come your way. Understanding the steps involved in solving equations like this helps build a strong foundation for more advanced mathematical concepts. So, let's begin!

Step 1: Eliminating Fractions – Multiplying by the Least Common Multiple

The first step in solving the equation $\frac{3}{4}x + \frac{9}{4} = \frac{1}{7}x - 5$ is to eliminate the fractions. Fractions can often complicate the process of solving equations, so removing them simplifies the equation and makes it easier to work with. To achieve this, we employ a technique that involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators. The denominators in our equation are 4 and 7. The least common multiple of 4 and 7 is 28. This is because 28 is the smallest number that is divisible by both 4 and 7. Now, we multiply both sides of the equation by 28. This ensures that the equality of the equation remains intact, as we are performing the same operation on both sides. When we multiply each term by 28, the fractions will cancel out, leaving us with an equation that contains only integers. Let's break down the multiplication step-by-step. First, we multiply the left side of the equation by 28: 28 * ($\frac{3}{4}x + \frac{9}{4}$) = 28 * $\frac{3}{4}x$ + 28 * $\frac{9}{4}$. Simplifying this, we get 21x + 63. Next, we multiply the right side of the equation by 28: 28 * ($\frac{1}{7}x - 5$) = 28 * $\frac{1}{7}x$ - 28 * 5. Simplifying this, we get 4x - 140. Therefore, after multiplying both sides of the original equation by 28, we obtain a new equation without fractions: 21x + 63 = 4x - 140. This new equation is much simpler to work with, as it only contains integers. In the next step, we will proceed to collect like terms, bringing all the 'x' terms to one side of the equation and all the constant terms to the other side. By eliminating fractions in this initial step, we have paved the way for a more streamlined and efficient solution process. This technique of multiplying by the LCM is a crucial skill in solving equations involving fractions, and mastering it will significantly enhance your ability to tackle more complex problems. So, remember to always look for opportunities to eliminate fractions when solving equations, as it often simplifies the process considerably.

Step 2: Collecting Like Terms – Isolating the Variable

After successfully eliminating fractions in the previous step, our equation now stands as 21x + 63 = 4x - 140. The next crucial step in solving for 'x' is to collect like terms. This involves grouping the terms containing the variable 'x' on one side of the equation and the constant terms on the other side. This strategic arrangement brings us closer to isolating the variable, which is our ultimate goal. To begin, let's focus on moving the 'x' terms to one side. We can achieve this by subtracting 4x from both sides of the equation. Subtracting 4x from both sides maintains the equality, as we are performing the same operation on both sides. This gives us: 21x - 4x + 63 = 4x - 4x - 140. Simplifying this, we get 17x + 63 = -140. Now, we have successfully grouped the 'x' terms on the left side of the equation. The next step is to move the constant terms to the other side. We can do this by subtracting 63 from both sides of the equation. Again, this maintains the balance of the equation. Subtracting 63 from both sides gives us: 17x + 63 - 63 = -140 - 63. Simplifying this, we get 17x = -203. Now, we have successfully collected all the 'x' terms on the left side and all the constant terms on the right side. The equation is now in a much simpler form, making it easier to isolate the variable 'x'. We are just one step away from finding the solution. In the next step, we will divide both sides of the equation by the coefficient of 'x' to finally isolate the variable and determine its value. The process of collecting like terms is a fundamental technique in solving equations. It allows us to organize the equation in a way that makes it easier to isolate the variable. By strategically adding or subtracting terms from both sides of the equation, we can manipulate the equation to our advantage and bring ourselves closer to the solution. Remember to always perform the same operation on both sides of the equation to maintain the equality. Mastering this technique is essential for solving a wide range of algebraic equations.

Step 3: Isolating the Variable – Division to Find the Solution

Following the successful collection of like terms, our equation now reads 17x = -203. The final step in solving for 'x' is to isolate the variable completely. This involves eliminating the coefficient that is multiplying 'x'. In our case, the coefficient is 17. To isolate 'x', we perform the inverse operation of multiplication, which is division. We divide both sides of the equation by 17. Dividing both sides by the same number maintains the equality of the equation. This gives us: $\frac{17x}{17} = \frac{-203}{17}$. Simplifying this, we get x = -11.94 (approximately -12). Therefore, the solution to the equation $\frac{3}{4}x + \frac{9}{4} = \frac{1}{7}x - 5$ is x = -11.94. We have successfully isolated the variable 'x' and determined its value. This value of 'x' satisfies the original equation, meaning that when we substitute -11.94 for 'x' in the original equation, both sides of the equation will be equal. The process of isolating the variable by division is a fundamental step in solving many types of equations, not just linear equations. It is a powerful technique that allows us to unravel the value of the unknown variable. By dividing both sides of the equation by the coefficient of the variable, we effectively undo the multiplication and reveal the value of the variable. Remember to always divide both sides of the equation by the same number to maintain the equality. This ensures that the solution we obtain is accurate. In this case, dividing both sides by 17 allowed us to isolate 'x' and find its value. The ability to isolate the variable is a crucial skill in algebra. It is the key to solving equations and unlocking the mysteries of mathematical relationships. By mastering this technique, you will be well-equipped to tackle a wide range of mathematical problems. So, remember the power of division in isolating variables, and you will be well on your way to becoming a proficient equation solver.

Step 4: Verification – Ensuring the Accuracy of the Solution

Having arrived at the solution x = -11.94 for the equation $\frac{3}{4}x + \frac{9}{4} = \frac{1}{7}x - 5$, it is crucial to verify its accuracy. Verification is a vital step in the problem-solving process, as it ensures that the solution we have obtained is correct and satisfies the original equation. To verify our solution, we substitute the value of x = -11.94 back into the original equation. This means replacing every instance of 'x' in the equation with -11.94. If the left-hand side (LHS) of the equation equals the right-hand side (RHS) after the substitution, then our solution is correct. Let's perform the substitution: $\frac{3}{4}(-11.94) + \frac{9}{4} = \frac{1}{7}(-11.94) - 5$. Now, we simplify both sides of the equation separately. On the left-hand side: $\frac{3}{4}(-11.94) + \frac{9}{4} = -8.955 + 2.25 = -6.705$. On the right-hand side: $\frac{1}{7}(-11.94) - 5 = -1.7057 - 5 = -6.7057$. Comparing the LHS and RHS, we see that they are approximately equal (-6.705 ≈ -6.7057). The slight difference is due to rounding during the calculation. This close agreement between the LHS and RHS strongly suggests that our solution of x = -11.94 is correct. Verification is not just a formality; it is a safeguard against errors. It helps us catch any mistakes we may have made during the solving process, such as incorrect arithmetic or algebraic manipulations. By verifying our solution, we can have confidence in our answer and avoid submitting an incorrect solution. In addition to catching errors, verification also deepens our understanding of the equation and its solution. It reinforces the concept that the solution is a value that makes the equation true. By substituting the solution back into the equation, we are essentially confirming this fundamental principle. So, remember to always verify your solutions, especially in exams or assessments. It is a small investment of time that can yield significant benefits in terms of accuracy and confidence. Verification is the final seal of approval on your solution, ensuring that you have successfully solved the equation.

Conclusion: Mastering the Art of Solving Equations

In this comprehensive guide, we have meticulously walked through the process of solving the linear equation $\frac{3}{4}x + \frac{9}{4} = \frac{1}{7}x - 5$. We began by understanding the fundamental principles of solving linear equations, emphasizing the importance of isolating the variable. We then tackled the equation step-by-step, starting with eliminating fractions by multiplying by the least common multiple, followed by collecting like terms to simplify the equation. We then isolated the variable through division, arriving at the solution x = -11.94. Finally, we verified the accuracy of our solution by substituting it back into the original equation. This step-by-step approach not only helped us find the solution but also reinforced the underlying concepts and techniques involved in solving linear equations. The ability to solve equations is a cornerstone of mathematical proficiency. It is a skill that is not only essential for academic success but also for solving real-world problems. From calculating budgets to designing structures, equations are used extensively in various fields. By mastering the art of solving equations, you equip yourself with a powerful tool that can be applied in diverse contexts. The key to success in solving equations lies in understanding the underlying principles and practicing consistently. Each step in the process, from eliminating fractions to isolating the variable, requires careful attention to detail and a solid grasp of algebraic manipulations. By working through numerous examples and challenging yourself with different types of equations, you can hone your skills and build confidence. Remember that mathematics is not just about memorizing formulas; it is about understanding concepts and applying them creatively. Solving equations is a testament to this principle. It requires you to think critically, analyze the equation, and choose the appropriate techniques to arrive at the solution. So, embrace the challenge of solving equations, and view it as an opportunity to develop your mathematical prowess. With dedication and practice, you can master this art and unlock the power of mathematics. The journey of mathematical discovery is a rewarding one, and solving equations is a significant milestone along the way.